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B.TECH ADMISSION IN INDIABy:
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LECTURE 14: LAPLACE TRANSFORM PROPERTIES
5 Laplace transform (3 lectures): Laplace transform as Fourier transform with
convergence factor. Properties of the Laplace transform
Specific objectives for today:• Linearity and time shift properties• Convolution property• Time domain differentiation & integration
property• Transforms table
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LECTURE 14: RESOURCES
Core material SaS, O&W, Chapter 9.5&9.6
Recommended material MIT, Lecture 18
Laplace transform properties are very similar to the properties of a Fourier transform, when s=j
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REMINDER: LAPLACE TRANSFORMS Equivalent to the Fourier transform when s=j
Associated region of convergence for which the integral is finite
Used to understand the frequency characteristics of a signal (system)
Used to solve ODEs because of their convenient calculus and convolution properties (today)
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dtetxsX st)()(
)()( sXtxL
j
j
stdsesXj
tx
)(
2
1)(
Laplace transform
Inverse Laplace transform
LINEARITY OF THE LAPLACE TRANSFORM
If
and
Then
This follows directly from the definition of the Laplace transform (as the integral operator is linear). It is easily extended to a linear combination of an arbitrary number of signals
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)()( 11 sXtxL
)()( 22 sXtxL
)()()()( 2121 sbXsaXtbxtaxL
ROC=R1
ROC=R2
ROC= R1R2
TIME SHIFTING & LAPLACE TRANSFORMS If
Then
Proof Now replacing t by t-t0
Recognising this as
A signal which is shifted in time may have both the magnitude and the phase of the Laplace transform altered.
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)()( sXtxL
)()( 00 sXettx st
L
j
j
stj dsesXtx
)()( 21
)()}({ 00 sXettxL st
j
j
ststj
j
j
ttsj
dsesXe
dsesXttx
))((
)()(
0
0
21
)(21
0
ROC=R
ROC=R
EXAMPLE: LINEAR AND TIME SHIFT Consider the signal (linear sum of two time shifted sinusoids)
where x1(t) = sin(0t)u(t). Using the sin() Laplace
transform example
Then using the linearity and time shift Laplace transform properties
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)4(5.0)5.2(2)( 11 txtxtx
0}Re{)(20
20
1
ss
sX
0}Re{5.02)(20
2045.2
s
seesX ss
CONVOLUTION The Laplace transform also has the multiplication property, i.e.
Proof is “identical” to the Fourier transform convolution
Note that pole-zero cancellation may occur between H(s) and X(s) which extends the ROC
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ROC=R1
ROC=R2
ROCR1R2
)()( sXtxL
)()( sHthL
)()()(*)( sHsXthtx
L
}{1)()(
1}{1
2)(
2}{2
1)(
ssHsX
ss
ssH
ss
ssX
EXAMPLE 1: 1ST ORDER INPUT & FIRST ORDER SYSTEM IMPULSE RESPONSE
Consider the Laplace transform of the output of a first order system when the input is an exponential (decay?)
Taking Laplace transforms
Laplace transform of the output is
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asas
sX
}Re{1
)(
)()(
)()(
tueth
tuetxbt
at
bsbs
sH
}Re{,1
)(
},max{}Re{11
)( basbsas
sY
Solved with Fourier transforms when a,b>0
EXAMPLE 1: CONTINUED … Splitting into partial fractions
and using the inverse Laplace transform
Note that this is the same as was obtained earlier, expect it is valid for all a & b, i.e. we can use the Laplace transforms to solve ODEs of LTI systems, using the system’s impulse response
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},max{}Re{111
)( basbsasab
sY
)()()( 1 tuetuety btatab
)()( sHthL
EXAMPLE 2: SINUSOIDAL INPUT Consider the 1st order (possible unstable) system response with input x(t)
Taking Laplace transforms
The Laplace transform of the output of the system is therefore
and the inverse Laplace transform is
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)()cos()(
)()(
0 tuttx
tueth at
asas
sH
}Re{1
)(
0}Re{)(20
2
ss
ssX
asa
a
s
as
a
asass
ssY
11
},0max{}Re{1
)(
20
220
2
20
20
2
20
2
ataettaa
tuty
)cos()sin(
)()( 0002
02
DIFFERENTIATION IN THE TIME DOMAIN Consider the Laplace transform derivative in the
time domain
sX(s) has an extra zero at 0, and may cancel out a corresponding pole of X(s), so ROC may be larger
Widely used to solve when the system is described by LTI differential equations
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ROC=R)()( sXtxL
j
j
stdsesXj
tx
)(
2
1)(
)()(
ssXdt
tdx L
ROCR
j
j
stdsessXjdt
tdx
)(
2
1)(
EXAMPLE: SYSTEM IMPULSE RESPONSE Consider trying to find the system response (potentially
unstable) for a second order system with an impulse input x(t)=(t), y(t)=h(t)
Taking Laplace transforms of both sides and using the linearity property
where r1 and r2 are distinct roots, and calculating the inverse transform
The general solution to a second order system can be expressed as the sum of two complex (possibly real) exponentials
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)()()()(
2
2
txtcydt
tdyb
dt
tyda
)()())((
11)()}({
1))}(({
)}({)()()(
2
2
1
1
212
2
2
2
rs
k
rs
k
rsrsacbsassHtyL
cbsastyL
tLtycLdt
tdybL
dt
tydaL
)()()( 2121 tuektuekty trtr
LECTURE 14: SUMMARY Like the Fourier transform, the Laplace transform is linear and represents time shifts (t-T) by multiplying by e-sT
Convolution
Convolution in the time domain is equivalent to multiplying the Laplace transforms
Laplace transform of the system’s impulse response is very important H(s) = h(t)e-stdt. Known as the transfer function.
Differentiation
Very important for solving ordinary differential equations
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)()()(*)( sHsXthtxL
)()(
ssXdt
tdx L
ROCR
ROCR1R2
QUESTIONS Theory SaS, O&W, Q9.29-9.32 Work through slide 12 for the first order system
Where the aim is to calculate the Laplace transform of the impulse response as well as the actual impulse response
Matlab Implement the systems on slides 10 & 12 in
Simulink and verify their responses by exact calculation.
Note that roots() is a Matlab function that will calculate the roots of a polynomial expression
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)()()(
ttbydt
tdya
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